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Ampere’s Law, etc. Text 30.3 - 30.5 • Ampere’s Law examples • Magnetic Flux and Gauss’s Law for magnetism Practice: Chapter 30, problems 35, 37, 39, 47, 57. Ampère’s Law 1 Review quiz: The integral in Ampere’s Law must be taken A) over any closed surface B) along a current-carrying wire in any closed circuit C) along any current-carrying wire D) along any closed curve Calculating B from Ampere’s Law For a few simple shapes, we can calculate the field B easily from Ampere’s Law. The steps: i) use symmetry to deduce the general behaviour ii) pick a closed curve on which to apply Ampere’s Law, taking advantage of the symmetry iii) Calculate (in terms of |B|) iv) Calculate Iencircled v) Apply Ampere’s Law to get B 2 Quick Quizzes What is the magnetic field inside and outside 1) a hollow copper pipe carrying current I? 2) a coaxial cable carrying equal currents (in opposite directions) on the inner and outer conductors? Single Coil: B I Several Coils: x x x x x 3 d l a I out of page b c B xxxxxxxxxxxxxxxx I into page Outside: B≈ 0 Inside: B is uniform, parallel to solenoid axis d c l aI b B xxxxxxxxxxxxxxxx On which sides of the green rectangle is not zero? A) bc and ad B) bc C) ad D) ab, bc, and cd E) none of the above is completely correct 4 l d a I b c B xxxxxxxxxxxxxxxx The “current encircled” by the green rectangle shown is A) equal to I B) equal to 6I C) equal to zero d c l a I out of page b B xxxxxxxxxxxxxxxx I into page Outside: B≈ 0 Inside: B = µ0nI (uniform, parallel to solenoid axis) where n = number of turns per metre of length 5 Example: I = 1 amp, n = 100 turns/cm (1x104 turns/m) Quiz: What is the field at the end of the solenoid, on the axis ? A) 130 gauss B) 65 gauss C) 260 gauss D) zero B r B path of integration for Show that, inside the toroid, (N = total number of turns) 6 Flux through a surface S: ( dA is the “area vector”, perpendicular to the surface.) - a scalar ; units, 1 T·m2 = 1 weber (Wb) - represents “number of field lines through S” The magnetic flux through any closed surface is zero. This is equivalent to: - the field lines form closed loops - there are no isolated magnetic poles (“magnetic monopoles”) 7 + + + + + + + + + + + + A beam of fast protons is also a current. Electric forces cause the beam to widen. The magnetic forces tend to A) cause the beam to spread even more B) cause the beam to spread less C) have no effect on the spreading What about a beam of fast electrons? At what speed v is the total force zero? FE v + FM R + v “current element” Ι Δs = qv 8